Statistical mechanics of addition polymerisation. Calculations of the expectation and variance of the average atmosphere in growing self avoiding walks

Recent calculations on the change in radial dimensions of reacting (growing) polyethylene in the gas phase experiencing Lennard Jones and Kihara type potentials revealed that a single reacting polyethylene molecule does not experience polymer collapse. This implies that a transition that is the converse of what happens when molten polyethylene crystallizes, i.e. it transforms from random coil like structure to folded rigid rod type structure, occurs. The predicted behaviour of growing polyethylene was explained by treating the head of the growing polymer chain as myopic whereas as the whole chain (i.e. when under equilibrium conditions) being treated as having normal vision, i.e. the growing chain does not see the attractive part of the LJ or Kihara Potentials. In this paper we provide further proof for this argument in two ways. Firstly we carry forward the exact enumeration calculations on growing self avoiding walks reported in that paper to larger values of number of steps by using Monte Carlo type calculations. We thereby assign physical significance to the connective constant of self avoiding walks, which until now was treated as a purely abstract mathematical entity. Secondly since a reacting polymer molecule that grows by addition polymerisation sees only one step ahead at a time, we extend this calculation by estimating the average atmosphere for molecules, with repulsive potential only (growing self avoiding walks in two dimensions), that look at two, three, four, five ...steps ahead. Our calculation shows that the arguments used in the previous work are correct.